# designFracDelayFIR

## Syntax

## Description

designs a fractional
delay FIR filter of length 50 and delay 0.5. The output `h`

= designFracDelayFIR`h`

is the vector
of filter coefficients.

specifies options using one or more name-value arguments.`h`

= designFracDelayFIR(`Name=Value`

)* (since R2024a)*

For example,

designs a fractional delay FIR filter of delay 0.3. The function automatically determines
the appropriate FIR length `h`

=
`designFracDelayFIR`

(`FractionalDelay`

=0.3,`Bandwidth`

=0.6)*N* for which the combined bandwidth is at least
0.6. As the `SystemObject`

argument is `true`

, the
function returns a `dsp.FIRFilter`

object.

When you specify only a partial list of filter parameters, the function designs the filter by setting the other design parameters to their default values.

## Examples

### Design Fractional Delay FIR Filter by Using Delay and Filter Length

Design a fractional delay FIR filter using the `designFracDelayFIR `

function. Pass the delay and the filter length as the input arguments to the function. Vary the filter length and observe the effect on the measured combined bandwidth and the nominal group delay.

**Vary Filter Length**

Set delay to 0.25 and filter length to 8 and design the fractional delay FIR filter.

fd = 0.25; [h1,i10,bw1] = designFracDelayFIR(FractionalDelay=fd,FilterLength=8)

`h1 = `*1×8*
-0.0086 0.0417 -0.1355 0.8793 0.2931 -0.0968 0.0341 -0.0074

i10 = 3

bw1 = 0.5810

The nominal group delay of the filter `i10+fd`

equals 3.25 samples. The measured combined bandwidth of the filter is 0.5810 in normalized frequency units.

Repeat the process with a filter length of 32 taps.

[h2,i20,bw2] = designFracDelayFIR(FractionalDelay=fd,FilterLength=32)

`h2 = `*1×32*
-0.0001 0.0004 -0.0009 0.0017 -0.0029 0.0046 -0.0071 0.0104 -0.0148 0.0208 -0.0291 0.0410 -0.0594 0.0926 -0.1752 0.8983 0.2994 -0.1252 0.0758 -0.0515 0.0367 -0.0266 0.0193 -0.0139 0.0098 -0.0067 0.0044 -0.0028 0.0016 -0.0009 0.0004 -0.0001

i20 = 15

bw2 = 0.8571

The nominal group delay of the filter now equals 15.25 samples. By increasing the filter length, the integer latency `i0`

increases, resulting in an increase in the nominal group delay. The combined bandwidth of the filter also increases to 0.8571 in normalized frequency units.

Increase the filter length to 64 taps. The group delay increases to 31.25 samples, and the integer latency is 31 samples. The measured combined bandwidth of the filter further increases to 0.9219, covering 92.19% of the overall bandwidth. As the filter length continues to increase, the combined bandwidth tends closer towards 1.

[h3,i30,bw3] = designFracDelayFIR(FractionalDelay=fd,FilterLength=64)

`h3 = `*1×64*
-0.0000 0.0001 -0.0001 0.0002 -0.0003 0.0004 -0.0006 0.0008 -0.0010 0.0013 -0.0017 0.0022 -0.0027 0.0034 -0.0042 0.0051 -0.0061 0.0074 -0.0088 0.0105 -0.0125 0.0149 -0.0177 0.0212 -0.0255 0.0311 -0.0386 0.0494 -0.0664 0.0979 -0.1787 0.8997 0.2999 -0.1277 0.0801 -0.0575 0.0442 -0.0352 0.0288 -0.0239 0.0200 -0.0168 0.0142 -0.0120 0.0101 -0.0085 0.0071 -0.0059 0.0049 -0.0040

i30 = 31

bw3 = 0.9219

**Plot Magnitude Response**

Plot the magnitude response of the three filters. Mark the measured combined bandwidth (MBW) of the three filters. By increasing the filter length, you can see that the measured combined bandwidth increases.

[H1,w] = freqz(h1,1); H2 = freqz(h2,1); H3 = freqz(h3,1); figure; plot(w/pi,mag2db(abs([H1 H2 H3]))) hold on hline = lines; xline(bw1, LineStyle = '--', LineWidth = 2, Color = hline(1,:)) xline(bw2, LineStyle = '--', LineWidth = 2, Color = hline(2,:)) xline(bw3, LineStyle = '--', LineWidth = 2, Color = hline(3,:)) hold off title('Magnitude Responses in dB') xlabel("Normalized Frequency (\times\pi rad/sample)") ylabel("Magnitude (dB)") grid legend('N = 8','N = 32','N = 64',... 'MBW (N = 8)',... 'MBW (N = 32)',... 'MBW (N = 64)',Location='southwest')

**Plot Group Delay Response**

Plot the group delay response of the three filters. Mark the nominal group delay *i0* + *fd* of the three filters. By increasing the filter length, you can see that the nominal group delay increases.

[g1,w] = grpdelay(h1,1); g2 = grpdelay(h2,1); g3 = grpdelay(h3,1); figure; plot(w/pi,[g1 g2 g3]) hline = lines; yline(i10+fd, LineStyle = '--', LineWidth = 2, Color = hline(1,:)) yline(i20+fd, LineStyle = '--', LineWidth = 2, Color = hline(2,:)) yline(i30+fd, LineStyle = '--', LineWidth = 2, Color = hline(3,:)) title('Group Delay Responses') xlabel("Normalized Frequency (\times\pi rad/sample)") ylabel("Group Delay") grid legend('N = 8','N = 32','N = 64',... 'Nominal Group Delay (N = 8)',... 'Nominal Group Delay (N = 32)',... 'Nominal Group Delay (N = 64)',Location='southwest'); ylim([-60,40]);

### Design Fractional Delay FIR Filter by Using Delay and Combined Bandwidth

Design a fractional delay FIR filter using the `designFracDelayFIR `

function. Pass the delay and the combined bandwidth as input arguments to the function.

Set delay to 0.786 and the target combined bandwidth to be 0.8. The function designs a filter that has a length of 22 taps, an integer latency *i0* of 10 samples, and a combined bandwidth `m`

*bw* of 0.8044 in normalized frequency units. This m*bw* value results in a combined bandwidth coverage of 80.44% of the frequency domain and exceeds the specified target combined bandwidth. The nominal group delay of the filter * i0+fd* equals 10.786.

fd = 0.786; tbw = 0.8; [h,i0,mbw] = designFracDelayFIR(FractionalDelay=fd,Bandwidth=tbw)

`h = `*1×22*
0.0003 -0.0011 0.0026 -0.0052 0.0094 -0.0156 0.0248 -0.0386 0.0611 -0.1052 0.2512 0.9225 -0.1548 0.0769 -0.0455 0.0281 -0.0173 0.0102 -0.0057 0.0028 -0.0012 0.0003

i0 = 10

mbw = 0.8044

Plot the impulse response of the FIR.

stem((0:length(h)-1),h); xlabel('h'); ylabel('h[n]'); title('Impulse Response of the Fractional Delay FIR')

Plot the resulting magnitude response and the group delay response. Mark the nominal group delay and the combined bandwidth of the filter.

[H1,w] = freqz(h,1); G1 = grpdelay(h,1); figure; yyaxis left plot(w/pi,mag2db(abs(H1))) ylabel("Magnitude (dB)") hold on yyaxis right plot(w/pi,G1) ylabel("Group Delay (in samples)") hline = lines; xline(mbw, LineStyle =':', Color = 'b', LineWidth = 2) xline(tbw, LineStyle = '--', Color = 'm', LineWidth = 2) yline(i0+fd, LineStyle = ':', Color = 'r', LineWidth = 1) yticks([i0, i0+fd,i0+1:i0+9]); hold off title('Magnitude Responses (dB) and Group Delay', FontSize = 10) xlabel("Normalized Frequency (\times\pi rad/sample)") legend('Gain Response','Group Delay Response','Measured Combined Bandwidth',... 'Target Combined Bandwidth','Nominal Group Delay', ... Location = 'west', FontSize = 10)

### Design Fractional Delay FIR Filter and Compare with Shifted Input

Design a fractional delay FIR filter using the `designFracDelayFIR`

function. Determine the group delay of the designed filter. Create a `dsp.FIRFilter`

object that uses these designed coefficients and hence has the same group delay. Alternately, create a sampled sequence of a known function. Pass the sampled sequence to the FIR filter. Compare the output of the FIR filter to the shifted samples of the known function. Specify this shift to be equal to the group delay of the FIR filter. Verify that the two sequences match.

Set the delay of the FIR filter to 1/3 and the length to 6 taps.

fd = 1/3; len = 6;

Design the filter using the `designFracDelayFIR`

function and determine the center index *i0* and the combined bandwidth `bw`

of the filter. The group delay of this filter is `i0`

+ `fd`

or approximately 2.33 for the bandwidth of `bw`

.

[h,i0,bw] = designFracDelayFIR(FractionalDelay=fd,FilterLength=len)

`h = `*1×6*
0.0293 -0.1360 0.7932 0.3966 -0.1088 0.0257

i0 = 2

bw = 0.5158

Create a `dsp.FIRFilter`

object and set its numerator to the filter coefficients *h*. This filter is now effectively a fractional delay FIR filter. Verify that the group delay response of this filter is approximately 2.33 for the duration of the bandwidth *bw*.

fdf = dsp.FIRFilter(h); grpdelay(fdf)

**Compare with Shifted Function**

Define a sequence *x as *samples of a known function.

```
f = @(t) (0.1*t.^2+cos(0.9*t)).*exp(-0.1*(t-5).^2);
n = (0:19)'; t = linspace(0,19,512);
x = f(n); % Samples
```

Plot the sampled values *x* against the original known function f(*t*).

scatter(n,x,20,'k','filled'); hold on; plot(t,f(t),'color',[0.5 0.5 0.5],'LineWidth',0.5) hold off; xlabel('Time') legend(["Input Samples","f(t)"]) title('Input Sequence with Known Underlying Analog Signal') ax = gca; ax.XGrid='on';

Pass the sampled sequence *x* through the FIR filter. Plot the input sequence and output sequence.

y = fdf(x); subplot(2,1,1); stem(x); title('Input Sequence'); xlabel('n') subplot(2,1,2) stem(y); title('FIR Output Sequence'); xlabel('n')

Shift the input sequence horizontally by `i0`

+ `fd`

, which is equal to the group delay of the FIR filter. Plot the function f(*t-i0-FD*). Verify that the input and output sequences fall roughly on the shifted function.

figure scatter(n,y,20,'red','filled') hold on; scatter(n+i0+fd,x,20,'black','filled') plot(t,f(t-i0-fd),'Color',[1,0.5,0.5],'LineWidth',0.1) xlabel('Time') legend(["Filter output","Shifted Input Samples","Shifted f(t-i0-fd)"]) hold off grid on title('Input and Output Sequences Aligned and Overlaid')

## Input Arguments

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

**Example: **`designFracDelayFIR(FractionalDelay=0.4,Bandwidth=0.8,SystemObject=true)`

designs and returns a `dsp.FIRFilter`

object.

`FractionalDelay`

— Fractional delay of filter

`0.5`

| positive scalar in the range [0,1]

Fractional delay of the filter, specified as a positive scalar in the range [0,1].
The fractional delay value that you specify determines the measured combined bandwidth
`MBW`

of the filter.

When you set `FractionalDelay`

to `0`

or
`1`

, the designed filter has a full bandwidth.

**Data Types: **`single`

| `double`

`FilterLength`

— Length of FIR filter

`50`

| integer greater than 1

Length of the fractional delay FIR filter, specified as an integer greater than 1.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`Bandwidth`

— Target combined bandwidth

positive scalar less than 0.999

Target combined bandwidth, specified as a positive scalar less than 0.999. This is the value of the combined bandwidth that the function must satisfy. Combined bandwidth is defined as the minimum of the gain bandwidth and the group delay bandwidth.

Specifying both filter length and target combined bandwidth results in an overdetermined design. The function does not support specifying both the values. When you specify the target combined bandwidth, the function determines the corresponding filter length, and designs the filter accordingly.

Specify a higher target combined bandwidth for a longer filter. For example,
setting `Bandwidth`

to 0.9 yields a filter of length of 52, while
increasing `Bandwidth`

to 0.99 yields a length of 724, which is
more than 10 times longer. As `Bandwidth`

tends towards 1, the
filter length theoretically tends towards infinity.

**Data Types: **`single`

| `double`

`SystemObject`

— Option to create filter System object™

`false`

(default) | `true`

*Since R2024a*

Option to create a filter System object, specified as one of these:

`false`

–– The function returns a vector of fractional delay FIR filter coefficients.`true`

–– The function returns a`dsp.FIRFilter`

System object.

**Data Types: **`logical`

`Verbose`

— Option to print function call in MATLAB^{®}

`false`

(default) | `true`

Option to print the entire function call in MATLAB, specified as one of these:

`false`

–– The function does not print the function call.`true`

–– The function prints the entire function call including the default values of the`Name=Value`

arguments that you did not specify when calling the function.Use this argument to view all the values used by the function to design and implement the filter.

**Data Types: **`logical`

## Output Arguments

`h`

— Designed filter

row vector | `dsp.FIRFilter`

object

`dsp.FIRFilter`

Designed filter, returned as one of these options.

Fractional delay FIR filter coefficients –– The function returns a row vector of length

*N*when you set the`SystemObject`

argument to`false`

.You can specify

*N*through the`FilterLength`

argument. If you specify the`Bandwidth`

argument, the function treats this value as the desired combined bandwidth, determines the corresponding filter length, and designs the filter accordingly.When you use filter length to design the filter, and you specify single-precision values in any of the input arguments, the function outputs single-precision filter coefficients.

*(since R2024a)*Multirate FIR filter object –– The function returns a

`dsp.FIRFilter`

object when you set the`SystemObject`

argument to`true`

.

**Data Types: **`single`

| `double`

`i0`

— Integer latency

integer

Integer latency of the designed FIR filter, returned as an integer value. Integer
latency is the smallest integer shift required to make the symmetric Kaiser window
causal. This value is approximately equal to half the filter length or
*N*/2. For more details, see Integer latency, i0.

The nominal group delay of the filter is given by
`i`

+_{0}*fd*, where
*fd* is the value you specify through the
`FractionalDelay`

argument.

**Data Types: **`single`

| `double`

`MBW`

— Measured combined bandwidth

positive scalar less than `0.999`

Measured combined bandwidth, returned as a real positive scalar less than
`0.999`

. This is the value of the combined bandwidth of the designed
filter. Combined bandwidth is defined as the minimum of gain bandwidth and
group delay
bandwidth.

The function designs the filter such that the measured combined bandwidth
`MBW`

meets or exceeds the target combined bandwidth. The function
determines the filter length so that it meets the bandwidth constraint.

When you specify the filter length *N*, the function treats this
value as the desired length of the filter. The measured combined bandwidth in this case
varies with the length. Larger the value of *N*, higher is the measured
combined bandwidth `MBW`

. This plot shows the variation. As the
filter length increases, the combined bandwidth of the filter moves closer towards 1.
The red dashed vertical line marks the combined bandwidth for each length. The
fractional delay value for each of these filters is 0.3

**Data Types: **`single`

| `double`

## More About

### Fractional Delay FIR Filter

The fractional delay FIR filter is an FIR approximation of an ideal
sinc shift filter with a fractional (noninteger) delay value *fd* in the
range [0,1].

The ideal shift filter models a band-limited D/A interpolator followed by shifted A/D uniform sampling. Assuming a uniform sampling rate and shift invariant interpolation, the resulting overall system can be expressed as a convolution filter, approximated by an FIR filter. In other words, $$y[n]={h}_{fd}[n]\ast x[n]$$, which encapsulates the D/A interpolation, shift, and A/D sampling chain as depicted in the figure.

where,

$$\begin{array}{l}{h}_{fd}[n]=\mathrm{sinc}(n-fd),\\ \widehat{x}(t)={\displaystyle \sum _{k}x[k]\mathrm{sinc}(t-k)}\\ \Rightarrow \widehat{x}(t+fd)={\displaystyle \sum _{k}x[k]\mathrm{sinc}}\left(t+fd-k\right)\end{array}$$

The frequency response of the ideal shift filter is

$${H}_{fd}(\omega )={e}^{-j\omega fd}.$$

The ideal shift filter has a flat unity gain response, and a constant
group delay of *fd*, where *fd* is the fractional delay
value you specify through the `FractionalDelay`

argument.

The function computes the FIR approximation by truncating the ideal filter and weighting the truncated filter by a Kaiser window.

$$\widehat{x}(n+fd)\approx y[n]=\left(h\ast x\right)\left[n\right],\text{\hspace{1em}}where\text{\hspace{1em}}h[m]=\mathrm{sinc}(m+fd)\xb7{K}_{N,\beta}[m]$$

where, $${K}_{N,\beta}[m]$$ is a Kaiser window of length *N* and has a shape
parameter β. The function designs the Kaiser window to optimize the FIR frequency response,
maximizing the combined bandwidths of both gain response and group delay response.

To make the FIR approximation causal, the algorithm introduces an additional shift of
`i`

, making the nominal group delay of the
filter equal to _{0}`i`

+_{0}*fd*. The
frequency response of the truncated filter is $$H(\omega )={e}^{-j\omega \left(fd+{i}_{0}\right)}$$.

For more details, see Integer latency, i0.

### Integer latency, `i`_{0}

_{0}

Integer latency, `i`

, is the
smallest integer shift that is required to make the symmetric Kaiser window causal._{0}

The ideal `sinc`

shift filter is an allpass filter, which has an
infinite and noncausal impulse response. To approximate this filter, the function uses a
finite index Kaiser window of length `N`

that is symmetric around the
origin and captures the main lobe of the `sinc`

function.

Due to the symmetric nature of the window, half of the window (approximately equal to
*N*/2) is on the negative side of the origin making the truncated filter
anti-causal. To make the truncated filter causal, shift the anti-causal (negative indices)
part of the FIR window by an integer latency,
`i`

, that is approximately equal to
_{0}*N*/2.

The overall delay of the causal FIR filter is
`i`

+_{0}*fd*, where
*fd* is the fractional delay.

For more details on FIR approximation, see the Causal FIR Approximations of an Ideal sinc Shift Filter section in Design Fractional Delay FIR Filters.

### Gain Bandwidth

Given an FIR frequency response *H(ω)*, the gain
bandwidth is the largest interval [0 *Ba*] over which the gain response
|*H(ω)*| is close to 1 up to a given tolerance value,
*tol*.

$${B}_{a}=\underset{\omega}{\mathrm{max}}\left\{\left|\left|H(\nu )\right|-1\right|<tol\text{\hspace{0.17em}}\forall \text{\hspace{0.17em}}0\le \nu \le \omega \right\}$$

### Group Delay Bandwidth

Given tolerance *tol* and group delay response
*G*, the group delay bandwidth is the largest interval [0
*Bg*] such that the group delay is close to the nominal value
*fd*.

$${B}_{g}=\underset{\omega}{\mathrm{max}}\left\{\left|G(\nu )-fd-{i}_{0}\right|<tol\text{\hspace{0.17em}}\forall \text{\hspace{0.17em}}0\le \nu \le \omega \right\}$$

### Combined Bandwidth

Combined bandwidth is defined as the minimum between gain bandwidth and group delay bandwidth.

$${B}_{c}=\mathrm{min}({B}_{a},{B}_{g})$$

Combined bandwidth depends on the fractional delay *fd* and the length
of the FIR filter *N*.

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

When you set the `SystemObject`

argument to `false`

,
the function supports code generation with no limitations.

When you set the `SystemObject`

argument to `true`

, the inputs to the function must be constants when
generating code.* (since R2024a)*

This function supports strict single precision in
generated code. When you use filter length to design the filter, and you specify any of the
input arguments in single-precision, the code you generate uses strictly single-precision
arithmetic.* (since R2024a)*

## Version History

**Introduced in R2021a**

### R2024a: `Value`

only syntax is discouraged

Starting in R2024a, specifying the arguments using the `Value`

only
syntax is discouraged in the `designFractionalDelayFIR`

function.

Existing instances of the function using the `Value`

only arguments
continue to run but are discouraged. Instead, specify the arguments using the
`Name=Value`

syntax.

### R2024a: Support for `Name=Value`

syntax

Starting in R2024a, the `designFractionalDelayFIR`

function supports
specifying the input arguments using the `Name=Value`

syntax.

Here is the table that shows how to replace your existing code.

Existing code | Replace with |
---|---|

`designFractionalDelayFIR` (fd) | `designFractionalDelayFIR` (`FractionalDelay` =fd) |

| `designFractionalDelayFIR` (`FractionalDelay` =fd,`FilterLength` =N) |

| `designFractionalDelayFIR` (`FractionalDelay` =fd,`Bandwidth` =TBW) |

### R2024a: Support for strict single precision

When you use filter length to design the filter, and you specify any of the input arguments in single-precision, the function designs filter coefficients in single precision both in simulation and in generated code.

### R2024a: New `'SystemObject'`

argument

You can now generate a `dsp.FIRFilter`

object from the
`designFracDelayFIR`

function by specifying the
`'SystemObject'`

argument to `true`

.

### R2024a: Support for code generation when `'SystemObject'`

flag is `true`

The `designFractionalDelayFIR`

function supports code generation when
you set the `'SystemObject'`

flag to `true`

.

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